Problem 21 - Algebra 2

Question

Find the LCM of each set of polynomials. $$ x^{2}-y^{2}, x^{3}+x^{2} y $$

Step-by-Step Solution

Answer

The LCM is \(x^2(x + y)(x - y)\).

Step 1: Factor the First Polynomial

Start by factoring the polynomial \(x^2 - y^2\). This is a difference of squares, which can be factored into \((x - y)(x + y)\). So, \(x^2 - y^2 = (x - y)(x + y)\).

Step 2: Factor the Second Polynomial

Now factor the polynomial \(x^3 + x^2y\). Notice that you can factor \(x^2\) out of both terms: \(x^3 + x^2y = x^2(x + y)\).

Step 3: Identify Unique Factors

List all unique factors from both polynomials. From \(x^2 - y^2\), the factors are \(x - y\) and \(x + y\). From \(x^3 + x^2y\), the factors are \(x^2\) and \(x + y\). Unique factors combined are \(x - y\), \(x + y\), and \(x^2\).

Step 4: Construct the LCM

The LCM is constructed by taking the highest power of each unique factor: \((x - y)^1\), \((x + y)^1\), and \(x^2\). Thus, the LCM is \(x^2(x + y)(x - y)\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a technique used to break down complex expressions into simpler components. It transforms a given polynomial into a product of its factors. Understanding how to factor polynomials is a crucial skill in algebra because it helps in simplifying equations and solving them.

When you factor a polynomial, you're essentially expressing it in terms of multiples that, when multiplied together, produce the original polynomial. For instance, if you have a polynomial like \(x^3 + x^2y\), you can take out the common factor of \(x^2\), resulting in \(x^2(x + y)\). This process reveals structures and simplifies further operations, such as finding the Least Common Multiple (LCM) of polynomials in our problem.

In our exercise, notice how the factoring process helps us parse down complex parts into more manageable pieces. This sets the stage for identifying unique factors, which are critical in determining the LCM.

Difference of Squares

The difference of squares is a specific polynomial form that can be factored easily. It occurs when you have a subtraction between two terms, both of which are perfect squares. Generally, the expression \(a^2 - b^2\) can be factored as \((a - b)(a + b)\). This is an essential tool in simplifying polynomials and appears often in algebra.

In the problem provided, \(x^2 - y^2\) is an example of the difference of squares. Each term \(x^2\) and \(y^2\) are perfect squares, which allows us to factor the entire expression into \((x - y)(x + y)\). Recognizing this pattern helps us express the polynomial in a factored form.

This method not only simplifies the polynomial itself but makes it easier to integrate its factors into more comprehensive calculations, such as determining the LCM with other polynomials.

Polynomial Factorization

Polynomial factorization entails breaking down a polynomial into a product of its factors or simpler polynomials. This process is key when dealing with multiple polynomials, as it allows us to examine and work with the smallest building blocks of algebraic expressions.

In the context of finding the Least Common Multiple (LCM), factorization is used to identify all unique elements from each polynomial's construction. By finding the greatest power of each factor across provided polynomials, you can determine the LCM. In the original solution, these steps of factorization were employed to find the common factors across \(x^2 - y^2\) and \(x^3 + x^2y\).

By listing factors like \(x - y\), \(x + y\), and \(x^2\), the process highlights which elements need to be taken into account for constructing the final LCM equation. Proper polynomial factorization is indispensable for solving polynomial equations effectively, making it an important aspect of algebra.